Non-failure of Filaments and Global Existence for the Equations of Fiber Spinning

In this work we give a global-in-time existence and uniqueness proof for solutions of the isothermal equations of fiber spinning. Fiber spinning is a widely used manufacturing process for the production of long thin filaments. In this process a highly viscous fluid is withdrawn from a reservoir and stretched to form a long fiber. The equations modeling fiber spinning are essentially based on cross-sectional averaging of the axisymmetric Stokes equations with free boundary. They form a coupled system consisting of a nonlinear mass transport equation and a nonlinear momentum conservation equation for dominant viscous forces in one space dimension. Our analytical approach is based on a representation result of the fiber cross-sectional area in terms of a suitably chosen Lagrangian variable. This representation shows that viscous fibers do not break in finite time and that solutions retain their smoothness. These two results suffice to extend local-in-time solutions to global-in-time solutions. Our no-breakup result is in agreement with previous related work.